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Week 2, Long Day: Polar coordinates and Exponential Form

Just as we are getting familiar with the rectangular form of complex numbers we will now introduce yet another way of describing and performing calculations with complex numbers: polar coordinates and the geometric rules and methods that follow. We will then continue from polar coordinates to the exponential form of complex numbers. It is essential to become good at using and switching between the different forms and to understand their respective advantages. At the foundation of polar coordinates and the exponential form we find the trigonometric functions cosine and sine. We will use the opportunity to recap on these function and become fully familiar with them.

Today’s Key Concepts

Angles in degrees and radians. The trigonometric functions cosine and sine. Modulus. Argument and principal argument. Polar coordinates. Geometric rules of calculations. The complex exponential function. Exponential form of complex numbers. Conversion techniques from rectangular to exponential form.

Preparation and Syllabus

Todays topics are from eNote 1 Complex Numbers, sections 1.5 to 1.8.

Dynamic Geometry: GeoGebra

Often in the Autumn semester we will use the geometry software GeoGebra to emphasize important points in the subject matter. Today we will use the software in exercise 5 to visualize geometric operations with complex numbers. You can download GeoGebra free of charge on your computer from this link or you can use a browser version.

Activity Program

  • 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: Your teachers are present in the study areas

Group Exercises

  1. Today’s Wetware Exercise
  2. Angular Measures in Degrees and Radians
  3. Cosine and Sine with the Unit Circle
  4. Polar Coordinates versus Rectangular Form
  5. Geometric Addition and Multiplication
  6. The Exponential Form of Complex Numbers
  7. The Complex Exponential Function
  8. Double Angles (Advanced)

If you want a printable version of the exercises without answers, go directly to your browser’s print function before expanding any hint or answer tabs.